# Talk till you're stuck, Spring 2020

This semester we will try something different. Please check back later or email me to sign up for the mailing list this semester.

Before attending this seminar, make sure to email me so you don't show up for nothing (in case there is no meeting this week).

Tentatively, I am going to have the talks on Monday, 4:10 -- 5:25 PM. Schedule of talks.

1. Day, Date, time, room, speaker.
2. Monday, Feb 3, 4:10-5:25PM, 507, Joaquin Moraga
3. Monday, Feb 10, 4:10-5:25PM, 507, Will Chen
4. Monday, Feb 17, 4:10-5:25PM, 507, Carl Lian
5. Monday, Feb 24, 4:10-5:25PM, 507, Will Sawin
6. Monday, March 2, 4:10-5:25PM, 507, Dmitrii Pirozhkov
7. Monday, March 9, 4:10-5:25PM, 507, NO MEETING
8. Monday, March 16, Spring Recess
9. Monday, March 23, 4:10-5:25PM, 507, Semon Rezchikov
10. Monday, March 30, 4:10-5:25PM, TBA, Daniel Litt
11. Monday, April 6, 4:10-5:25PM, TBA, TBA
12. Monday, April 13, 4:10-5:25PM, TBA, David Hamann
13. Monday, April 20, 4:10-5:25PM, TBA, TBA
14. Monday, April 27, 4:10-5:25PM, TBA, TBA
15. Monday, May 4, 4:10-5:25PM, TBA, TBA

Abstracts.

1. Joaquin Moraga. Title: On the Jordan property for local fundamental groups. Abstract: In this talk, we discuss the Jordan property for the local fundamental group of klt singularities. We also show how the existence of a large Abelian subgroup of such a group reflects on the geometry of the singularity. We show a characterization theorem for klt 3-fold singularities with large local fundamental groups. Finally, we will discuss possible generalizations to higher dimensions.
2. Will Chen. Title: Strong approximation for markoff equations. Abstract: The markoff equation is the affine cubic surface x^2 + y^2 + z^2 = xyz. This equation has quite a long history, and has come up historically in the study of diophantine approximation and quadratic forms, but it also has connections with hyperbolic geometry. In this talk I'll describe a relatively recent conjecture of Bourgain, Gamburd, and Sarnak, which asserts that for every prime p, the set of integer solutions surjects onto the set of F_p solutions. It turns out this conjecture can be phrased in terms of the connectedness of a moduli space of elliptic curves equipped with certain nonabelian level structures. These moduli spaces are algebraic curves, noncongruence modular curves, and also "teichmuller curves". Using this latter perspective, I will describe how this conjecture can be phrased in terms of lengths of closed geodesics on certain (congruence) modular curves equipped with a flat geometry. If true, this conjecture (and its generalizations) would give interesting new arithmetic/geometric information about the corresponding noncongruence modular curves, as well as giving some new information about the tame fundamental group of P^1 minus three points in characteristic p. Bourgain-Gamburd-Sarnak establish this conjecture for all but a thin (but infinite) set of primes p, but their method is a direct analytical attack on the equation itself. It would be nice if a complete solution could be attained using these geometric perspectives. I will try to keep this talk elementary.
3. Carl Lian. Title: Beyond the Alexander-Hirschowitz Theorem. Abstract: The Alexander-Hirschowitz Thoerem, proven in the 1990's, states that, for n fixed, the linear system of degree d hypersurfaces in P^n singular at a collection of r general points has the expected dimension except for a finite, explicit list of (d,r). A dream is that the same is true for an arbitrary polarized variety. The proof for projective spaces proceeds by degenerating the points into special position and applying induction, but this does not seem to be a viable strategy in the general case, as I will indicate. An alternative, direct approach that works for P^2 can be extended to arbitrary surfaces with some effort, but applying the same method to threefolds leads to some innocuous-looking geometric questions I don't know how to solve. I will explain this circle of ideas; hecklers welcome.
4. Will Sawin. Title: The space of nilpotent Higgs fields on a fixed vector bundle. Abstract: Higgs fields on a vector bundle V on a curve C are global homomorphisms from V to V tensor K_C. We call them nilpotent if, at a generic point, locally trivializing K_C, they are nilpotent endomorphisms. Higgs fields on V form an affine space, and nilpotent Higgs fields a closed subscheme. I would like to give a description, as nice as possible, of the set of vector bundles where this subscheme has a large-dimensional irreducible component. I would also like to describe the geometry of this component. I have some partial results in the case of rank 2 vector bundles. This is all motivated by problems in analytic number theory.
5. Dmitrii Pirozhkov. Title: Coherent sheaves on punctured spaces and their extensions. Abstract: Let U be the complement of the origin in an affine space A^n. Given a coherent sheaf F on U, can it be extended over the puncture to a coherent sheaf on the entire A^n, and if it can be, in how many (interesting) ways? There are many variants of this question: maybe there is some group action involved, or maybe we consider complexes of coherent sheaves, etc. I have come across a certain question like that while thinking about semiorthogonal decompositions on projective spaces. There are many basic aspects which I don’t understand, so I hope that the audience can help me with them!
6. Semon Rezchikov. Title: Title: Holomorphic aspects of Floer homology Abstract: To a symplectic manifold M, Floer theory associates the Donaldson-Fukaya category Fuk(M) with objects (approximately) the Lagrangian submanifolds of M and morphisms the Lagrangian Floer homology groups. If M is hyperkahler then the Fukaya category of M with respect to a *real* symplectic form is conjectured to be computable in terms of pure algebraic geometry. This construction appears however to miss some of the structure symplectic geometry provides: it looks like the Hom spaces in this category should be the fibers of vector bundles underlying certain D-modules over P^1. I will review the more established parts of this story and then will attempt to explain why I think this extra structure should exist by discussing some examples. I have many confusions about the Riemann-Hilbert correspondence for irregular holonomic D-modules and about the geometry of holomorphic Lagrangians in general, so maybe you all can teach me some basic algebraic geometry.
7. Daniel Litt. Title: Ceresa classes and independence of l. Abstract: Let X be a variety over a finitely generated field k; one expects that the Galois representations on the l-adic cohomology groups
H^i(X_{\bar k}, Q_l)
are, in a suitable sense, independent of l. Similarly, given a cycle Z of codimension d on X, one expects that its image under the cycle class map
CH^d(X) → H^{2d}(X, Q_l(d))
lives in a piece of a certain arithmetically defined filtration, again independent of l in a suitable sense (this is part of the Beilinson-Bloch-Murre conjectures). I'll discuss the relationship between these two conjectures in a very special case, and some partial results (e.g. over p-adic fields) for certain very special cycles on Jacobians and products of curves -- namely the so-called Ceresa and modified diagonal cycles. One interesting feature of this special case is that the Galois representations under consideration are not semisimple. This is joint work with Wanlin Li, and it'll be fun to talk about where we're stuck.
8. David Hamann. Title: On the topology of p-adic Bun_{G}. Abstract: Let X = P^1, G a reductive group over C, and Bun_{G} the moduli space of G-bundles on X. Starting from Grothendieck's classification of vector bundles on P^1_{C}, one sees that points of the underlying topological space |Bun_{G}| are (essentially) parametrized in terms of dominant cocharacters of G, denoted X_*(G)^{dom}, where the cocharacter is recording the slope polygon of a G-bundle and each point corresponds to a unique Harder-Narasimhan strata inside Bun_{G}. Using extensions and modifications of G-bundles, one can topologize this parametrization, by endowing X_*(G)^{dom} with the topology coming from the partial ordering given by the Bruhat order. Similarly, one can replace X with the Fargues-Fontaine curve and G with a p-adic group, and then one has a similar parametrization of the analogous space |Bun_{G}| (essentially) in terms of rational cocharacters (X_*(G) \otimes \mathbb{Q})^{dom} or more precisely the Kottwitz set of G due to Fargues. It is a conjecture of Fargues that one can similarly topologize this parametrization. For G = GL_{n}, this is a theorem of Hansen et al. For more general reductive groups, this is current work in progress of the author. In this talk, we will sketch the strategy of proof in the more pathological case of X = P^1_{C}, following some ideas of Braverman, Gaitsgory, and Scheider on Drinfeld's Compactifications. At the end, we will see that everything reduces to a concrete statement about bundles that is easy to verify for X = P^1, but less straight-forward to verify for the Fargues-Fontaine curve due to the existence of rational slopes. We will then explain how to work it out for some classical groups, and try to discuss how one might approach the general case.